As previously stated by EnergyNumbers, ML is very often used in financial and economics forecasting, just like OLS. The deep learning is less useful in economics academia because you do not know the cause and it will be hard to publish a well-explained paper based on that.
What kind of methods/algorithms/models, if any, are frequently used for what kind of purposes among what kind of economists? Where can I see their work (e.g. blog, paper, etc...)
You could just use goole scholar and search for "financial time series forecasting machine learning". GARCH, SVM, and ARIMA are often used. You could get started here:
Kim, Kyoung-jae. "Financial time series forecasting using support vector machines." Neurocomputing 55.1 (2003): 307-319.
Hsu, Ming-Wei, et al. "Bridging the divide in financial market forecasting: machine learners vs. financial economists." Expert Systems with Applications 61 (2016): 215-234.
Shynkevich, Yauheniya, et al. "Forecasting stock price directional movements using technical indicators: investigating window size effects on one-step-ahead forecasting." Computational Intelligence for Financial Engineering & Economics (CIFEr), 2104 IEEE Conference on. IEEE, 2014.
Tan, Elliot. "Forecasting Foreign Exchange Rates in Response to Federal Reserve Communication: A Machine Learning Approach." (2016).